Elementary proof. Quite possibly there are few notions as
important as this for the philosophy and history of mathematics, as I
shall explain. [Elementary proof as a tool of Felix Klein's Plan A rather
than Plan B mathematics; and elementary proof as a perfectly achieved
local foundations.]
My starting point was the essay of a leading mathematician,
Gian-Carlo Rota, his essay on mathematical truth, Chapter 9 of Indiscrete
Thoughts (Birkhauser Boston, 1997).
Harvey Friedman's suggestion that one use "self-contained proof"
instead of "elementary proof" is actually historically and philosophically
apt, as I will explain. At the same time, I worry about his claim about
how "elementary" is used by 20th century mathematicians. There is not only
Rota's use. -- But the literature I've been following out in algebra and
number theory, most recent, where declarations of the existence or
nonexistence of elementary proof occur, it does seem that what is at issue
is whether or not a more or less maximally "self-contained" proof has been
given [unless there is an esoteric side to this literature].
It is interesting as we are learning from FOM reports that there is
possibly considerable discrepancy among assertions in the literature about
whether or not elementary proofs exist for a theorem. Does this reveal
something about the difficulty of keeping track of the literature or
controversy about what counts as an elementary proof?
Rota argues that what is salient about a an elementary proof is
that it reveals a theorem to be "trivially true", true by the light of the
very terms that contain it, analytically true. This is not to say that
the proof is simple or easy to understand; and it is not to say that the
theorem is revealed to trivial in the sense of insignificant. Nor is
there the expectation that an elementary proof would give us the most
fruitful mathematical understanding of the theorem. It is rather perhaps
that one has a definitive proof, as it were a local foundations for that
bit of mathematics. [Once one has the definitive proof, then one is free
to devote one's energies to searching for proofs that lead to new ideas and
insights.]
[Warning on "analyticity" and alternative logics: there are two
senses of meaning containment, the beans in a jar sense of direct
containment -- Locke -- or "logically contained" -- Frege -- and it is
something like Frege's notion which is at issue here, except some
accommodation would have to be made for the thought that the "logic" must
be adjusted to the subject matter, and it must be that minimally supported
by the subject matter, perhaps in the leading sense in which Hilbert in On
the Infinite spoke of finitary arithmetic as having a logic distinct from
classical logic, but that he -- Hilbert -- couldn't be bothered to
articulate it, having bigger fish to fry. Maybe it is here that the two
sense of elementary proof connect!]
So the quest for elementary proofs is the quest for the most local,
minimal, self-contained, individual, isolating foundation for a
theorem, which at the same time is nonreductive, which tries, that is,
to take the wording or terms of the theorem at face value.
In Ele.Math from and AvcStandpoint:Algebra, Felix Klein
distinguishes Plan A mathematics and Plan B mathematics (there is also a
Plan C).
Plan A (if I don't have A & B reversed): separation and
self-containedness of mathematical subject matters.
Plan B: the unification of mathematics into one subject matter
(Naturphilosophie).
Plan A: Aristotelian
PlanB: Leibnizian.
Elementary proof as the basic tool of Plan A.
[Maybe the history of mathematics finds tendencies
oscillating between Plan A and Plan B, sort of like the life cycle of
slime molds.]
respectfully submitted,
rbrt tragesser